Question 1

$\text { Given } \mathbf { u } = \langle 1,3 \} \text { and } \mathbf { v } = \langle - 2,2 \} \text { find } \mathbf { u } \cdot \mathbf { v } \text {. }$

Accepted Answer

A) -8 
B) -4 
C) 4 
D) 7 
E) -1 
A) -8 
B) -4 
C) 4 
D) 7 
E) -1

Question 2

Identify the following quadric surface. $\frac { x ^ { 2 } } { 6 } - \frac { y ^ { 2 } } { 10 } - \frac { z ^ { 2 } } { 6 } = 1$

Accepted Answer

A)  ellipsoid 
B)  elliptic cone 
C)  elliptic paraboloid 
D)  hyperboloid of one sheet 
E)  hyperboloid of two sheets 
A)  ellipsoid 
B)  elliptic cone 
C)  elliptic paraboloid 
D)  hyperboloid of one sheet 
E)  hyperboloid of two sheets

Question 3

Find a unit vector u in the direction opposite of $$\mathbf { v } = \langle 2,8 , - 3 \}$$

Accepted Answer

A)  $\mathbf { u } = ( - 1 / \sqrt { 77 } ) ( 2,8 , - 3 )$ 
B)  $\mathbf { u } = ( 1 / \sqrt { 59 } ) ( 2,8 , - 3 )$ 
C)  $\mathbf { u } = ( - 1 / \sqrt { 59 } ) ( 2,8 , - 3 \}$ 
D)  $\mathbf { u } = ( 1 / \sqrt { 77 } ) \{ 2,8 , - 3 \}$ 
E)  $\mathbf { u } = ( - 1 / 6 \sqrt { 2 } ) \{ 2,8 , - 3 \}$ 
A)  $\mathbf { u } = ( - 1 / \sqrt { 77 } ) ( 2,8 , - 3 )$ 
B)  $\mathbf { u } = ( 1 / \sqrt { 59 } ) ( 2,8 , - 3 )$ 
C)  $\mathbf { u } = ( - 1 / \sqrt { 59 } ) ( 2,8 , - 3 \}$ 
D)  $\mathbf { u } = ( 1 / \sqrt { 77 } ) \{ 2,8 , - 3 \}$ 
E)  $\mathbf { u } = ( - 1 / 6 \sqrt { 2 } ) \{ 2,8 , - 3 \}$

Question 4

Find the equation of the surface satisfying the conditions, and identify the surface. The set of points equidistant from the point $( 2,4,5 )$ and the plane $y = - 1$.

Accepted Answer

A)  $10 y = ( x - 2 ) ^ { 2 } + ( z - 5 ) ^ { 2 } + 15$, an elliptic paraboloid 
B)  $( x - 2 ) ^ { 2 } + ( y - 4 ) ^ { 2 } + ( z - 5 ) ^ { 2 } - 5 = 0$, a sphere 
C)  $10 y = ( x - 2 ) ^ { 2 } + ( z - 5 ) ^ { 2 } + 1 ,$, a hyperbolic paraboloid 
D)  $( x - 2 ) ^ { 2 } + ( y - 4 ) ^ { 2 } + ( z - 5 ) ^ { 2 } - 1 = 0$, a sphere 
E)  $10 y = ( x - 2 ) ^ { 2 } + ( z - 5 ) ^ { 2 } - 1$, , a hyperbolic paraboloid 
A)  $10 y = ( x - 2 ) ^ { 2 } + ( z - 5 ) ^ { 2 } + 15$, an elliptic paraboloid 
B)  $( x - 2 ) ^ { 2 } + ( y - 4 ) ^ { 2 } + ( z - 5 ) ^ { 2 } - 5 = 0$, a sphere 
C)  $10 y = ( x - 2 ) ^ { 2 } + ( z - 5 ) ^ { 2 } + 1 ,$, a hyperbolic paraboloid 
D)  $( x - 2 ) ^ { 2 } + ( y - 4 ) ^ { 2 } + ( z - 5 ) ^ { 2 } - 1 = 0$, a sphere 
E)  $10 y = ( x - 2 ) ^ { 2 } + ( z - 5 ) ^ { 2 } - 1$, , a hyperbolic paraboloid

Question 5

Find the area of the triangle with the vertices $A = ( 0,0,0 ) , B = ( 3,1,2 ) , \text { and }$ $C = ( 1,1,2 ) .$ Round your answer to two decimal places.

Accepted Answer

A)  $20.00$ 
B)  $8.00$ 
C)  $6.00$ 
D)  $2.24$ 
E)  $4.47$ 
A)  $20.00$ 
B)  $8.00$ 
C)  $6.00$ 
D)  $2.24$ 
E)  $4.47$

Question 6

Given the vector v lies in the yz-plane, has magnitude 8, and makes $\text { angle of } 150 ^ { \circ }$ with the positive y-axis. Find the component form of v.

Accepted Answer

A)  $\mathbf { v } = \{ 0 , - 4 \sqrt { 3 } , \pm 4 \}$ 
B)  $\mathbf { v } = ( 4 \sqrt { 3 } , 0,4 \}$ 
C)  $\mathbf { v } = \{ 0 , - 4 \sqrt { 3 } , 4 \}$ 
D)  $\mathbf { v } = \{ - 4 \sqrt { 3 } , 0 , \pm 4 \}$ 
E)  $\mathbf { v } = \{ 0,4 \sqrt { 3 } , \pm 4 \}$ 
A)  $\mathbf { v } = \{ 0 , - 4 \sqrt { 3 } , \pm 4 \}$ 
B)  $\mathbf { v } = ( 4 \sqrt { 3 } , 0,4 \}$ 
C)  $\mathbf { v } = \{ 0 , - 4 \sqrt { 3 } , 4 \}$ 
D)  $\mathbf { v } = \{ - 4 \sqrt { 3 } , 0 , \pm 4 \}$ 
E)  $\mathbf { v } = \{ 0,4 \sqrt { 3 } , \pm 4 \}$

Question 7

$\text { Find the vector } \mathbf { z } = 4 \mathbf { u } + 4 \mathbf { v } - 3 \mathbf { w } \text {. }$ $\mathbf { v } = \{ 1 , - 2,5 \} , \mathbf { u } = \{ 6 , - 3 , - 6 \} , \mathbf { w } = \{ - 3,2 , - 5 )$

Accepted Answer

A)  $\mathbf { z } = ( - 37,26,11 )$ 
B)  $\mathbf { z } = ( 37 , - 26,11 )$ 
C)  $\mathbf { z } = ( - 37 , - 26,11 )$ 
D)  $\mathbf { z } = ( 37 , - 26 , - 11 )$ 
E)  $\mathbf { z } = ( 37,26,11 )$ 
A)  $\mathbf { z } = ( - 37,26,11 )$ 
B)  $\mathbf { z } = ( 37 , - 26,11 )$ 
C)  $\mathbf { z } = ( - 37 , - 26,11 )$ 
D)  $\mathbf { z } = ( 37 , - 26 , - 11 )$ 
E)  $\mathbf { z } = ( 37,26,11 )$

Question 8

Find the magnitude of the vector v given its initial and terminal points. Round your answer to four decimal places.
Initial point: (-2, -4, -5)
Terminal point: (-3, -9, -10)

Accepted Answer

A)  $\sqrt { 54 }$ or $7.3485$ 
B)  $\sqrt { 56 }$ or $7.4833$ 
C)  $\sqrt { 51 }$ or $7.1414$ 
D)  $\sqrt { 16 }$ or $4.0000$ 
E)  $\sqrt { 64 }$ or $8.0000$ 
A)  $\sqrt { 54 }$ or $7.3485$ 
B)  $\sqrt { 56 }$ or $7.4833$ 
C)  $\sqrt { 51 }$ or $7.1414$ 
D)  $\sqrt { 16 }$ or $4.0000$ 
E)  $\sqrt { 64 }$ or $8.0000$

Question 9

Find the area of the parallelogram with the vertices $A = ( 3,1,0 ) , B = ( 4,3,1 ) , C = ( 6,4,4 )$, and $D = ( 5,2,3 )$. Round your answer to two decimal places.

Accepted Answer

A)  $35.00$ 
B)  $3.00$ 
C)  $5.92$ 
D)  $7.00$ 
E)  $2.65$ 
A)  $35.00$ 
B)  $3.00$ 
C)  $5.92$ 
D)  $7.00$ 
E)  $2.65$

Question 10

Find  a set of symmetric equations of the line through the point $( 8,5,3 ) \text { parallel to }$ the vector $\mathrm { v } = \langle 2,4,4 \}$.

Accepted Answer

A)  $\frac { x - 8 } { - 3 } = \frac { z - 5 } { 4 } = \frac { y - 3 } { - 7 }$ 
B)  $\frac { x - 8 } { 2 } = \frac { y - 5 } { 4 } = \frac { z - 3 } { 4 }$ 
C)  $\frac { x - 8 } { 2 } = \frac { y + 5 } { 4 } = \frac { z + 3 } { 4 }$ 
D)  $\frac { x - 8 } { - 3 } = \frac { y - 5 } { 4 } = \frac { z - 3 } { - 7 }$ 
E)  $\frac { x - 8 } { 2 } = \frac { y + 5 } { 4 } = \frac { z - 3 } { 4 }$ 
A)  $\frac { x - 8 } { - 3 } = \frac { z - 5 } { 4 } = \frac { y - 3 } { - 7 }$ 
B)  $\frac { x - 8 } { 2 } = \frac { y - 5 } { 4 } = \frac { z - 3 } { 4 }$ 
C)  $\frac { x - 8 } { 2 } = \frac { y + 5 } { 4 } = \frac { z + 3 } { 4 }$ 
D)  $\frac { x - 8 } { - 3 } = \frac { y - 5 } { 4 } = \frac { z - 3 } { - 7 }$ 
E)  $\frac { x - 8 } { 2 } = \frac { y + 5 } { 4 } = \frac { z - 3 } { 4 }$

Question 11

Determine whether the lines given below intersect, and, if so, find the point of intersection. $$\begin{array} { l } 
\frac { x - 1 } { 5 } = \frac { y - 2 } { 7 } = \frac { z + 8 } { 5 } \
\frac { x - 3 } { 1 } = \frac { x - 15 } { - 2 } = \frac { z - 5 } { - 2 }
\end{array}$$

Accepted Answer

A)  The lines are parallel. 
B)  The lines intersect at $( 2,6 , - 33 )$. 
C)  The lines intersect at $( 7,11 , - 3 )$. 
D)  The lines intersect at $( 6,9 , - 1 )$. 
E)  The lines are not parallel but do not intersect. 
A)  The lines are parallel. 
B)  The lines intersect at $( 2,6 , - 33 )$. 
C)  The lines intersect at $( 7,11 , - 3 )$. 
D)  The lines intersect at $( 6,9 , - 1 )$. 
E)  The lines are not parallel but do not intersect.

Question 12

The vector v and its initial point is given. Find the terminal point. 
$\mathbf { v } = \{ 3 , - 2 \} , $ initial point $( - 6,10 )$

Accepted Answer

A)  $( - 3 , - 8 )$ 
B)  $( 8 , - 3 )$ 
C)  $( 3,8 )$ 
D)  $( 3 , - 8 )$ 
E)  $( - 3,8 )$ 
A)  $( - 3 , - 8 )$ 
B)  $( 8 , - 3 )$ 
C)  $( 3,8 )$ 
D)  $( 3 , - 8 )$ 
E)  $( - 3,8 )$

Question 13

Find an equation in rectangular coordinates for the equation given in cylindrical coordinates. $r ^ { 2 } + z ^ { 2 } = 4$

Accepted Answer

A)  $x + y + z = 2$ 
B)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ 
C)  $x ^ { 2 } + y ^ { 2 } + \left( \frac { z } { 2 } \right) ^ { 2 } = 0$ 
D)  $x ^ { 2 } + y ^ { 2 } = - 4 z ^ { 2 }$ 
E)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2$ 
A)  $x + y + z = 2$ 
B)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ 
C)  $x ^ { 2 } + y ^ { 2 } + \left( \frac { z } { 2 } \right) ^ { 2 } = 0$ 
D)  $x ^ { 2 } + y ^ { 2 } = - 4 z ^ { 2 }$ 
E)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2$

Question 14

Find the angle between the vectors for u and v given below. $$\mathbf { u } = \langle 1,1 \} , \mathbf { v } = \langle 3 , - 1 \}$$

Accepted Answer

A)  $63.43 ^ { \circ }$ 
B)  $24.09 ^ { \circ }$ 
C)  $26.57 ^ { \circ }$ 
D)  $69.62 ^ { \circ }$ 
E)  $84.26 ^ { \circ }$ 
A)  $63.43 ^ { \circ }$ 
B)  $24.09 ^ { \circ }$ 
C)  $26.57 ^ { \circ }$ 
D)  $69.62 ^ { \circ }$ 
E)  $84.26 ^ { \circ }$

Question 15

Find the area of the parallelogram that has the given vectors $\mathbf { u } = \mathbf { j } \text { and } \mathbf { v } = 2 \mathbf { j } + \mathbf { k } \text { as }$ adjacent sides.

Accepted Answer

A)  2 
B)  0 
C)  4 
D)  1 
E)  3 
A)  2 
B)  0 
C)  4 
D)  1 
E)  3

Question 16

Determine whether u and v are orthogonal, parallel or neither. $$\mathbf { u } = 18 \mathbf { i } + 6 \mathbf { j } , \mathbf { v } = 5 \mathbf { i } - 15 \mathbf { j }$$

Accepted Answer

A)  Parallel, same direction 
B)  Orthogonal 
C)  Neither parallel nor orthogonal 
D)  Parallel, opposite direction 
A)  Parallel, same direction 
B)  Orthogonal 
C)  Neither parallel nor orthogonal 
D)  Parallel, opposite direction

Question 17

$\text { Given } \mathbf { v } = \{ 1,6,6 \} \text {, find } \frac { 3 } { 2 } \mathbf { v } \text {. }$

Accepted Answer

A)  $\mathbf { v } = \left\langle \frac { 2 } { 3 } , 9,9 \right)$ 
B)  $\mathbf { v } = \left( \frac { 2 } { 3 } , 3,3 \right\}$ 
C)  $\mathbf { v } = \left( \frac { 3 } { 2 } , 9,9 \right)$ 
D)  $\mathbf { v } = \left( \frac { 3 } { 2 } , 3,3 \right)$ 
E)  $\mathbf { v } = \left( \frac { 3 } { 2 } , 6,6 \right\}$ 
A)  $\mathbf { v } = \left\langle \frac { 2 } { 3 } , 9,9 \right)$ 
B)  $\mathbf { v } = \left( \frac { 2 } { 3 } , 3,3 \right\}$ 
C)  $\mathbf { v } = \left( \frac { 3 } { 2 } , 9,9 \right)$ 
D)  $\mathbf { v } = \left( \frac { 3 } { 2 } , 3,3 \right)$ 
E)  $\mathbf { v } = \left( \frac { 3 } { 2 } , 6,6 \right\}$

Question 18

Given the vector v and its initial point find the terminal point of the vector. $\mathbf { v } = \langle 3,2,1 \} , \quad$ initial point $( 2 , - 4,5 )$

Accepted Answer

A)  $( 5,4 , - 6 )$ 
B)  $( 5 , - 2,6 )$ 
C)  $( - 5 , - 2,6 )$ 
D)  $( 5,4,6 )$ 
E)  $( - 5,4,6 )$ 
A)  $( 5,4 , - 6 )$ 
B)  $( 5 , - 2,6 )$ 
C)  $( - 5 , - 2,6 )$ 
D)  $( 5,4,6 )$ 
E)  $( - 5,4,6 )$

Question 19

Find an equation in rectangular coordinates for the equation given in cylindrical coordinates. $$r = 6 \sin \theta$$

Accepted Answer

A)  $x ^ { 2 } + y ^ { 2 } = 6 y$ 
B)  $x ^ { 2 } + y ^ { 2 } = 6 x$ 
C)  $x ^ { 2 } + y ^ { 2 } = \frac { x } { 6 }$ 
D)  $x ^ { 2 } + y ^ { 2 } = \frac { y } { 6 }$ 
E)  $x ^ { 2 } + z ^ { 2 } = 6 x$ 
A)  $x ^ { 2 } + y ^ { 2 } = 6 y$ 
B)  $x ^ { 2 } + y ^ { 2 } = 6 x$ 
C)  $x ^ { 2 } + y ^ { 2 } = \frac { x } { 6 }$ 
D)  $x ^ { 2 } + y ^ { 2 } = \frac { y } { 6 }$ 
E)  $x ^ { 2 } + z ^ { 2 } = 6 x$

Question 20

$\text { Given } \mathbf { u } = \mathbf { i } + 3 \mathbf { j } \text { and } \mathbf { v } = - 8 \mathbf { i } + \mathbf { j } + \mathbf { k } \text {, find } \mathbf { u } \times \mathbf { v } \text {. }$

Accepted Answer

A)  $6 \mathbf { i } - \mathbf { j } + 26 \mathbf { k }$ 
B)  $3 \mathbf { i } + \mathbf { j } + 24 \mathbf { k }$ 
C)  $3 \mathbf { i } - \mathbf { j } + 25 \mathbf { k }$ 
D)  $4 \mathbf { i } + \mathbf { j } - 25 \mathbf { k }$ 
E)  $3 \mathbf { i } - \mathbf { j } - 25 \mathbf { k }$ 
A)  $6 \mathbf { i } - \mathbf { j } + 26 \mathbf { k }$ 
B)  $3 \mathbf { i } + \mathbf { j } + 24 \mathbf { k }$ 
C)  $3 \mathbf { i } - \mathbf { j } + 25 \mathbf { k }$ 
D)  $4 \mathbf { i } + \mathbf { j } - 25 \mathbf { k }$ 
E)  $3 \mathbf { i } - \mathbf { j } - 25 \mathbf { k }$

$\text { Given } \mathbf { u } = \langle 1,3 \} \text { and } \mathbf { v } = \langle - 2,2 \} \text { find } \mathbf { u } \cdot \mathbf { v } \text {. }$

Identify the following quadric surface. $\frac { x ^ { 2 } } { 6 } - \frac { y ^ { 2 } } { 10 } - \frac { z ^ { 2 } } { 6 } = 1$

Find a unit vector u in the direction opposite of $\mathbf { v } = \langle 2,8 , - 3 \}$

Find the equation of the surface satisfying the conditions, and identify the surface. The set of points equidistant from the point $( 2,4,5 )$ and the plane $y = - 1$ .

Find the area of the triangle with the vertices $A = ( 0,0,0 ) , B = ( 3,1,2 ) , \text { and }$ $C = ( 1,1,2 ) .$ Round your answer to two decimal places.

Given the vector v lies in the yz-plane, has magnitude 8, and makes $\text { angle of } 150 ^ { \circ }$ with the positive y-axis. Find the component form of v.

$\text { Find the vector } \mathbf { z } = 4 \mathbf { u } + 4 \mathbf { v } - 3 \mathbf { w } \text {. }$ $\mathbf { v } = \{ 1 , - 2,5 \} , \mathbf { u } = \{ 6 , - 3 , - 6 \} , \mathbf { w } = \{ - 3,2 , - 5 )$

Find the magnitude of the vector v given its initial and terminal points. Round your answer to four decimal places. Initial point: (-2, -4, -5) Terminal point: (-3, -9, -10)

Find the area of the parallelogram with the vertices $A = ( 3,1,0 ) , B = ( 4,3,1 ) , C = ( 6,4,4 )$ , and $D = ( 5,2,3 )$ . Round your answer to two decimal places.

Find a set of symmetric equations of the line through the point $( 8,5,3 ) \text { parallel to }$ the vector $\mathrm { v } = \langle 2,4,4 \}$ .

Determine whether the lines given below intersect, and, if so, find the point of intersection. == ==

The vector v and its initial point is given. Find the terminal point. $\mathbf { v } = \{ 3 , - 2 \} ,$ initial point $( - 6,10 )$

Find an equation in rectangular coordinates for the equation given in cylindrical coordinates. $r ^ { 2 } + z ^ { 2 } = 4$

Find the angle between the vectors for u and v given below. $\mathbf { u } = \langle 1,1 \} , \mathbf { v } = \langle 3 , - 1 \}$

Find the area of the parallelogram that has the given vectors $\mathbf { u } = \mathbf { j } \text { and } \mathbf { v } = 2 \mathbf { j } + \mathbf { k } \text { as }$ adjacent sides.

Determine whether u and v are orthogonal, parallel or neither. $\mathbf { u } = 18 \mathbf { i } + 6 \mathbf { j } , \mathbf { v } = 5 \mathbf { i } - 15 \mathbf { j }$

$\text { Given } \mathbf { v } = \{ 1,6,6 \} \text {, find } \frac { 3 } { 2 } \mathbf { v } \text {. }$

Given the vector v and its initial point find the terminal point of the vector. $\mathbf { v } = \langle 3,2,1 \} , \quad$ initial point $( 2 , - 4,5 )$

Find an equation in rectangular coordinates for the equation given in cylindrical coordinates. $r = 6 \sin \theta$

$\text { Given } \mathbf { u } = \mathbf { i } + 3 \mathbf { j } \text { and } \mathbf { v } = - 8 \mathbf { i } + \mathbf { j } + \mathbf { k } \text {, find } \mathbf { u } \times \mathbf { v } \text {. }$

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

Exam 11: Vectors in the Plane

Given u=⟨1,3} and v=⟨−2,2} find u⋅v. \text { Given } \mathbf { u } = \langle 1,3 \} \text { and } \mathbf { v } = \langle - 2,2 \} \text { find } \mathbf { u } \cdot \mathbf { v } \text {. } Given u=⟨1,3} and v=⟨−2,2} find u⋅v.

Identify the following quadric surface. x26−y210−z26=1\frac { x ^ { 2 } } { 6 } - \frac { y ^ { 2 } } { 10 } - \frac { z ^ { 2 } } { 6 } = 16x2​−10y2​−6z2​=1

Find a unit vector u in the direction opposite of v=⟨2,8,−3}\mathbf { v } = \langle 2,8 , - 3 \}v=⟨2,8,−3}

Find the equation of the surface satisfying the conditions, and identify the surface. The set of points equidistant from the point (2,4,5)( 2,4,5 )(2,4,5) and the plane y=−1y = - 1y=−1 .

Find the area of the triangle with the vertices A=(0,0,0),B=(3,1,2), and A = ( 0,0,0 ) , B = ( 3,1,2 ) , \text { and }A=(0,0,0),B=(3,1,2), and C=(1,1,2).C = ( 1,1,2 ) .C=(1,1,2). Round your answer to two decimal places.

Given the vector v lies in the yz-plane, has magnitude 8, and makes angle of 150∘\text { angle of } 150 ^ { \circ } angle of 150∘ with the positive y-axis. Find the component form of v.

Find the magnitude of the vector v given its initial and terminal points. Round your answer to four decimal places. Initial point: (-2, -4, -5) Terminal point: (-3, -9, -10)

Find the area of the parallelogram with the vertices A=(3,1,0),B=(4,3,1),C=(6,4,4)A = ( 3,1,0 ) , B = ( 4,3,1 ) , C = ( 6,4,4 )A=(3,1,0),B=(4,3,1),C=(6,4,4) , and D=(5,2,3)D = ( 5,2,3 )D=(5,2,3) . Round your answer to two decimal places.

Find a set of symmetric equations of the line through the point (8,5,3) parallel to ( 8,5,3 ) \text { parallel to }(8,5,3) parallel to the vector v=⟨2,4,4}\mathrm { v } = \langle 2,4,4 \}v=⟨2,4,4} .

Determine whether the lines given below intersect, and, if so, find the point of intersection. == ==

The vector v and its initial point is given. Find the terminal point. v={3,−2},\mathbf { v } = \{ 3 , - 2 \} , v={3,−2}, initial point (−6,10)( - 6,10 )(−6,10)

Find an equation in rectangular coordinates for the equation given in cylindrical coordinates. r2+z2=4r ^ { 2 } + z ^ { 2 } = 4r2+z2=4

Find the angle between the vectors for u and v given below. u=⟨1,1},v=⟨3,−1}\mathbf { u } = \langle 1,1 \} , \mathbf { v } = \langle 3 , - 1 \}u=⟨1,1},v=⟨3,−1}

Find the area of the parallelogram that has the given vectors u=j and v=2j+k as \mathbf { u } = \mathbf { j } \text { and } \mathbf { v } = 2 \mathbf { j } + \mathbf { k } \text { as }u=j and v=2j+k as adjacent sides.

Determine whether u and v are orthogonal, parallel or neither. u=18i+6j,v=5i−15j\mathbf { u } = 18 \mathbf { i } + 6 \mathbf { j } , \mathbf { v } = 5 \mathbf { i } - 15 \mathbf { j }u=18i+6j,v=5i−15j

Given v={1,6,6}, find 32v. \text { Given } \mathbf { v } = \{ 1,6,6 \} \text {, find } \frac { 3 } { 2 } \mathbf { v } \text {. } Given v={1,6,6}, find 23​v.

Given the vector v and its initial point find the terminal point of the vector. v=⟨3,2,1},\mathbf { v } = \langle 3,2,1 \} , \quadv=⟨3,2,1}, initial point (2,−4,5)( 2 , - 4,5 )(2,−4,5)

Find an equation in rectangular coordinates for the equation given in cylindrical coordinates. r=6sin⁡θr = 6 \sin \thetar=6sinθ

Given u=i+3j and v=−8i+j+k, find u×v. \text { Given } \mathbf { u } = \mathbf { i } + 3 \mathbf { j } \text { and } \mathbf { v } = - 8 \mathbf { i } + \mathbf { j } + \mathbf { k } \text {, find } \mathbf { u } \times \mathbf { v } \text {. } Given u=i+3j and v=−8i+j+k, find u×v.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Exact First-Order Equations

Filters

$\text { Given } \mathbf { u } = \langle 1,3 \} \text { and } \mathbf { v } = \langle - 2,2 \} \text { find } \mathbf { u } \cdot \mathbf { v } \text {. }$

Identify the following quadric surface. $\frac { x ^ { 2 } } { 6 } - \frac { y ^ { 2 } } { 10 } - \frac { z ^ { 2 } } { 6 } = 1$

Find a unit vector u in the direction opposite of $\mathbf { v } = \langle 2,8 , - 3 \}$

Find the equation of the surface satisfying the conditions, and identify the surface. The set of points equidistant from the point $( 2,4,5 )$ and the plane $y = - 1$ .

Find the area of the triangle with the vertices $A = ( 0,0,0 ) , B = ( 3,1,2 ) , \text { and }$ $C = ( 1,1,2 ) .$ Round your answer to two decimal places.

Given the vector v lies in the yz-plane, has magnitude 8, and makes $\text { angle of } 150 ^ { \circ }$ with the positive y-axis. Find the component form of v.

Find the area of the parallelogram with the vertices $A = ( 3,1,0 ) , B = ( 4,3,1 ) , C = ( 6,4,4 )$ , and $D = ( 5,2,3 )$ . Round your answer to two decimal places.

Find a set of symmetric equations of the line through the point $( 8,5,3 ) \text { parallel to }$ the vector $\mathrm { v } = \langle 2,4,4 \}$ .

The vector v and its initial point is given. Find the terminal point. $\mathbf { v } = \{ 3 , - 2 \} ,$ initial point $( - 6,10 )$

Find an equation in rectangular coordinates for the equation given in cylindrical coordinates. $r ^ { 2 } + z ^ { 2 } = 4$

Find the angle between the vectors for u and v given below. $\mathbf { u } = \langle 1,1 \} , \mathbf { v } = \langle 3 , - 1 \}$

Find the area of the parallelogram that has the given vectors $\mathbf { u } = \mathbf { j } \text { and } \mathbf { v } = 2 \mathbf { j } + \mathbf { k } \text { as }$ adjacent sides.

Determine whether u and v are orthogonal, parallel or neither. $\mathbf { u } = 18 \mathbf { i } + 6 \mathbf { j } , \mathbf { v } = 5 \mathbf { i } - 15 \mathbf { j }$

$\text { Given } \mathbf { v } = \{ 1,6,6 \} \text {, find } \frac { 3 } { 2 } \mathbf { v } \text {. }$

Given the vector v and its initial point find the terminal point of the vector. $\mathbf { v } = \langle 3,2,1 \} , \quad$ initial point $( 2 , - 4,5 )$

Find an equation in rectangular coordinates for the equation given in cylindrical coordinates. $r = 6 \sin \theta$

$\text { Given } \mathbf { u } = \mathbf { i } + 3 \mathbf { j } \text { and } \mathbf { v } = - 8 \mathbf { i } + \mathbf { j } + \mathbf { k } \text {, find } \mathbf { u } \times \mathbf { v } \text {. }$